Extracting the Sheaf and espace étalé condition from an abstractly given equivalence between these two spaces I've been learning about sheaves and have, I think, understood its characterisation
as a space that glues together locally, and its presentation as a space over a base to which it is locally homeomorphic.
However I've come across the following description of this equivalence (Tom Leinster, Sheaves do not belong to algebraic geometry, http://www.maths.gla.ac.uk/~tl/sheaves.pdf) 
which is induced from an adjunction using general categorical ideas. His description doesn't mention either local homeomorphisms or the sheaf condition. Obviously they must be there, but I don't see how to extract it.
In brief: He uses the adjunction 
$Cat[A^{op},Set]$<->$B$
where $A$ is a small category, $B$ has small colimits, and we have
a functor $I:A \to B$, then 
$\leftarrow$ is $B[I,-]$ and
$\rightarrow$ is -(tensor)I which IS defined as the left adjoint, which he states
   exists as a particular colimit
He then looks at the restriction of the adjunction where it becomes an equivalence, then setting for an X in Top, $A:=Opn(X)$, $B:=Top/X$
and $I:A\to B=Opn(X) \to Top/X$ the obvious functor, then the adjunction
$PShf X=Cat[Opn(X)^{op},Set]$<->$Top/$X restricts to equivalence $Shf X$ <-> $Et X$
I Just wanted to note that a similar question has been posted on Math Overflow before:
Understanding the etale space construction from a formal viewpoint
and one of the answers suggests looking at Sketches of an Elephant: A Topos Theory Compendium by Johnstone, which is what I'm going to do
 A: The general definition of sheaf, is much more general than a local homeomorphism over a space- however, it is sheaves of this later kind that came about first. In the general framework, you have a small category $C$ and a $\textbf{Grothendieck topology}$ on $C$, which declares what families of arrows $\left(c_i \to c\right)$ are $\textbf{covering families}$. The sheaf condition for open covers has an obvious generalization in this setting. Sheaves form a reflective subcategory of presheaves, and the reflector, is given by the "sheafification" functor. In the classical setting of sheaves over a space, $C$ is the poset of open subsets of a space $X$, and the Grothendieck topology is given by open covers. There is a canonical functor $Open\left(X\right) \to Top/X$, and by general abstract nonsense, this produces an adjunction between $Psh\left(Open\left(X\right)\right)$ and $Top/X$. The essential image of $$Psh\left(Open\left(X\right)\right) \to Top/X$$ is those maps $Y \to X$ which are local homeomorphisms, and this functor is the "espace étalé" construction. The other functor, sends a map $Y \to X$ to its sheaf of sections. Starting with a presheaf $F$, constructing its espace étalé, and then taking it sections, yields a sheaf. This composition if a functor $Psh\left(Open\left(X\right)\right) \to Sh\left(X\right)$, and is in fact the sheafification functor. You can read more about this in, for instance, "Sheaves in Geometry and Logic".
